Bose-Einstein Condensates in Rotating Traps and Optical ... - BEC

Chapter 5
Groundstate of a BEC in an optical
lattice
We explore the groundstate of a condensate confined in a one-dimensional optical lattice within
GP-theory.
First, we consider a system which is uniform in the transverse direction (section 5.1,5.2 and
5.3). In this case, there are two tunable parameters: the lattice depth sER and the interaction
parameter gn where n is the average density. In the lattice direction the density profile is
strongly modulated by the periodic potential reflecting the concentration of the atoms at the
bottom of the potential wells. Energy and chemical potential are significantly increased with
respect to the uniform case, while the combined presence of lattice and repulsive interactions
between atoms leads to a decrease in the compressibility. The inverse compressibility has a
nonlinear dependence on average density since interactions counteract the compression of the
atoms by the external potential. In a deep lattice however, the equation of state of the system
takes the same linear dependence on density as in the uniform case and is charcterized by an
effective coupling constant ˜g which grows as a function of lattice depth.
The presence of repulsive interactions produces a screening effect since the particles resist
more to being squeezed in the potential wells than in the absence of interaction. We find
that increasing the interaction parameter gn corresponds to an effective decrease of the lattice
depth s.
In a second step, we allow for the additional presence of radial and axial harmonic trapping
(section 5.4). We use the results obtained for the purely periodic potential as an input to
calculate the groundstate properties in the combined trap. The scheme we develop adequately
describes current experimental settings in which the optical potential is superimposed to a
harmonically trapped TF-condensate leading to a distribution of atoms over many lattice sites.
We derive simple analytic expressions for the chemical potential and the density profile averaged
over each lattice period by using the effective coupling constant description. These expressions
allow us to explicitly calculate the occupation of each site and the radius of the cloud. We
discuss the dependence on lattice depth of the chemical potential, the condensate size and the
density at the center of the harmonic trap averaged over one lattice period.
Our results for the groundstate energy, chemical potential, compressibility and regarding
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